# Regular embedding

In algebraic geometry, a closed immersion of schemes is a regular embedding of codimension r if each point x in X has an open affine neighborhood U in Y such that the ideal of is generated by a regular sequence of length r.

For example, if X and Y are smooth over a scheme S and if i is an S-morphism, then i is a regular embedding. In particular, every section of a smooth morphism is a regular embedding. If is regularly embedded into a regular scheme, then B is a complete intersection ring.

The notion is used, for instance, in an essential way in Fulton's approach to intersection theory. The important fact is that when i is a regular embedding, if I is the ideal sheaf of X in Y, then the normal sheaf, the dual of , is locally free (thus a vector bundle) and the natural map is an isomorphism: the normal cone coincides with the normal bundle.

A flat morphism of finite type is called a (local) complete intersection morphism if each point x in X has an open affine neighborhood U so that f |U factors as where j is a regular embedding and g is smooth. For example, if f is a morphism between smooth varieties, then f factors as where the first map is the graph morphism and so is a complete intersection morphism.

## References

1. Sernesi, D. Notes 2.
2. Sernesi, D.1.
3. Sernesi, D.2.1.